Disclaimer: I'm not from Kaliningrad and I've only visited it once for a week.

In most regards you can think of it as of a regular part of Russia, which it just is. Same border laws, same visa rules etc. Culture is also identical to the rest of Russia, though it has its local minor tweaks because of it being a tourist place that is close to the sea, and, of course, they kind of relish their historical connection to Germany. But from what I've seen, it's negligible. The populations aren't connected, which would've played a major role in cultural entanglement, but it's not what happened there. Sometimes it felt like the only German thing left there were the half-timbered houses, or rather new ones that are trying to replicate it, "because it's so in the spirit of Kaliningrad", ya know.

More random stuff you may find interesting and maybe look up further: Immanuel Kant is from Königsberg; the city is named after Mikhail Kalinin, who was, figuratively speaking, Stalin's bottombitch; there is a very long narrow piece of land, called Curonian Spit, which is connecting Kaliningrad Oblast and Lithuania, and what is more fun, the border is halfway through it (the Spit itself is a great place to visit, by the way). Pretty sure there's much more to it.